Andrés Santos Universidad de Extremadura, Badajoz (Spain)

Andrés Santos Universidad de Extremadura, Badajoz (Spain)

Outline Moment equations molecules for Maxwell Some solvable states: Planar Fourier flow Planar Fourier flow with gravity Planar Couette flow Force-driven Poiseuille flow Uniform shear and longitudinal flows Conclusions 2

(1844-1906) (Cartoon by Bernhard Reischl, University of Vienna) 3

streaming external force collisions Differential cross section 4

Interaction potential 5

Weak form of the Boltzmann equation Density of ψ(v) Flux Source: external force Source: collisions 6

Hierarchy of moment equations 7

Maxwell molecules On the Dynamical Theory of Gases Phil. Trans. Roy. Soc. (London) 157, 49-88 (1867) In the present paper I propose to consider the molecules of a gas, not as elastic spheres of definite radius, but as small bodies or groups of smaller molecules repelling one another with a force whose direction always passes very nearly through the centres of gravity of the molecules, and whose magnitude is represented very nearly by some function of the distance of the centres of gravity. (1831-1879) I have made this modification of the theory in consequence of the results of my experiments on the viscosity of air at different temperatures, and I have deduced from these experiments that the repulsion is inversely as the fifth power of the distance. I have found by experiment that the coefficient of viscosity in a given gas is independent of the density, and proportional to the absolute temperature, so that if ET be the viscosity, ET p/ρ. 8

Maxwell molecules 9

Properties of the collisional moments in Maxwell models 10

Properties of the collisional moments in Maxwell models. Eigenfunctions and eigenvalues Eigenfunctions Eigenvalues 11

Properties of the eigenvalues in Maxwell models 12

Navier-Stokes constitutive equations Claude-Louis Navier (1785-1836) P ij = pδ ij η George Gabriel Stokes (1819-1903) µ i u j + j u i 2 3 uδ ij, η = p λ 02 13

1. Steady planar Fourier flow z=l T w ' Hot q z Jean-Baptiste Joseph Fourier (1768-1830) z=0 T w Cold 14

Asmolov, Makashev, and Nosik (1979) proved that an exact solution of the (nonlinear) Boltzmann equation for Maxwell molecules exists with 15

Dimensionless quantities (Local) Knudsen number: Reduced distribution function: Reduced moments: 16

Hierarchy of moment equations 17

j 6 5 4 3 2 1 0 2 4 6 8 10 12 14 k=2r+l 18

First few moments Fourier s law 19

First few moments 20

Benchmark for DSMC simulations Gallis et al., Phys. Fluids 18, 017104 (2006) 21

BGK description Same qualitative results for moments. Full velocity distribution function φ(c;²). Divergence of the (CE) expansion of φ(c;²) in powers of ². 22

2. Steady planar Fourier flow with gravity (Rayleigh-Bénard-like flow) Lord Rayleigh (1842-1919) z=l T w ' g>0 Hot Henri Bénard (1874-1939) q z z=0 T w Cold 23

Perturbation analysis Tij, Garzó, and Santos, Phys. Rev. E 56, 6729 (1997) akin to the Rayleigh number Pure Fourier flow 24

Corrections to Navier-Stokes 25

Corrections to Navier-Stokes Heating from below Heating from above 26

BGK description Same qualitative results. Moments up to order γ 6. Divergence of the expansion in powers of γ. 27

3. Steady planar Couette flow T=T w ' U w z=+l u x (z) Maurice Couette (1858-1943) z=-l -U w T=T -w 28

Makashev and Nosik (1981) proved that an exact solution of the (nonlinear) Boltzmann equation for Maxwell molecules exists with NS: θ(a)=1 29

Dimensionless quantities Knudsen numbers: Reduced distribution function: Reduced moments: 30

Hierarchy of moment equations 31

Corrections to Navier-Stokes 32

Perturbation analysis Tij and Santos, Phys. Fluids 7, 2857 (1995) super-burnett Burnett 33

Benchmark for DSMC simulations Gallis et al., Phys. Fluids 18, 017104 (2006) 34

BGK description Same qualitative results for moments. Explicit expressions for κ*(a), η*(a), θ(a), Φ(a), 1 (a), and 2 (a). Full velocity distribution function φ(c;²,a). Divergence of the expansion in powers of a. Influence of gravity analyzed. 35

4. Force-driven Poiseuille flow Jean-Louis-Marie Poiseuille (1797-1869) 36

Equal normal stresses Newton s law Fourier s law No longitudinal heat flux Temperature is maximal at the central layer (y=0)

Do NS predictions agree with computer simulations? (z) DSMC T 0 T max DSMC NS but... NS 2.5 mfp y max 38

A Burnett-order effect? DSMC In the slab y< y max, Burnett sgn q y = sgn ØT/Øy Heat flows from the colder to the hotter layers!! 39

Other Non-Newtonian properties Non-uniform pressure (z) (zz) Normal stress differences Longitudinal component of the heat flux (but no longitudinal thermal gradient!) 40

Hierarchy of moment equations 41

Perturbation analysis Tij, Sabanne, and Santos, Phys. Fluids 10, 1021 (1998) Equilibrium moments (at y=0) 42

Results to order g 2 : Hydrodynamic fields 43

Results to order g 2 : Hydrodynamic fluxes 44

Numerical values of the coefficients Coefficient Quantity NS Burnett a 13-moment b R13-moment c 19-moment d Exact C p p 0 1.2 1.2 1.2 1.2 1.2 C T T 0 0 0.56 0.9295 1.04 1.0153 C zz P zz 0 0 0 3.413? 6.4777 C yy P yy 0 0 0 3.36? 6.2602 C q q z 0 0.4 0.4 0.4 0.4 0.4 a Uribe & Garcia (1999) b Risso & Cordero (1998) c Taheri, Struchtrup & Torrilhon (2008) d Hess & Malek Mansour (1999) 45

1.04 9.56 1.02 T(y)/T 0 1.00 0.98 NS 11.6g 2 ρ 0 η 2 0 /p3 0 0.96-8 -6-4 -2 0 2 4 6 8 y/[η 0 (k B T 0 /m) 1/2 /p 0 ] 46

P ii (y)/p 0 1.10 1.08 1.06 1.04 zz xx 1.02 1.00 NS 6.48g 2 ρ 0 η 2 0 /p3 0 yy 6.26g 2 ρ 0 η 2 0 /p3 0 0.98-8 -6-4 -2 0 2 4 6 8 y/[η 0 (k B T 0 /m) 1/2 /p 0 ] 47

BGK description Same qualitative results. Hydrodynamic fields up to order g 5. Velocity distribution function up to order g 3. Divergence of the expansion in powers of g. Exact non-perturbative solution for a special value of g. 48

5. Other (quasi-uniform) states 49

In both cases, Exact rheological functions for arbitrary values (non-perturbative solution) of the corresponding Knudsen number (a=γ xy /λ 02, a=γ xx /λ 02 ). Divergence of the fourth-degree moments beyond a critical value a=a c. Algebraic high-velocity decay of the velocity distribution function for any a: Finite number of convergent moments. 50

Conclusions The hierarchy of moment equations can be recursively solved in the case of Maxwell molecules for some non-trivial states. No need to apply truncation closures. In some cases, a perturbation analysis is required. Exact results are important by themselves and also useful as an assessment of simulation techniques approximate methods kinetic models 51

More details about shear flows (including mixtures, BGK model description, ) Acknowlegments (in alphabetical order): J. Javier Brey, Universidad de Sevilla, Spain James. W. Dufty, University of Florida, USA Vicente Garzó, Universidad de Extremadura, Spain Mohamed Tij, Université Moulay Ismaïl, Morocco (Springer, 2003) THANKS FOR YOUR ATTENTION! 52